--- title: "Using a delay-adjusted case fatality ratio to estimate under-reporting" description: "Using a corrected case fatality ratio, we calculate estimates of the level of under-reporting for any country with greater than ten deaths" status: real-time-report rmarkdown_html_fragment: true update: 2020-07-24 authors: - id: tim_russell corresponding: true - id: joel_hellewell equal: 1 - id: sam_abbott equal: 1 - id: nick_golding - id: hamish_gibbs - id: chris_jarvis - id: kevin_vanzandvoort - id: ncov-group - id: stefan_flasche - id: roz_eggo - id: john_edmunds - id: adam_kucharski ---

Aim

To estimate the percentage of symptomatic COVID-19 cases reported in different countries using case fatality ratio estimates based on data from the ECDC, correcting for delays between confirmation-and-death.

Data availability

The under-reporting estimates for all countries can be downloaded as a single .csv file here.

Similarly, the prevalence estimates can be downloaded as a single .csv file here.

How to cite this work

If you wish to cite this work, please do cite the associated preprint [1]).

Methods Summary

The associated preprint[1], specifically the corresponding supplementary material contains a full description of the methods and limitations used to arrive at the estimates presented here.

Current estimates of under-reporting, prevalence and adjusted case curves along with reported cases

Temporal variation

Figure 1: Temporal variation in reporting rate. We calculate the percentage of symptomatic cases reported on each day a country has had more than ten deaths. We then fit a Gaussian Process (GP) to these data (see Temporal variation model fitting section for details), highlighting the temporal trend of each countries reporting rate. The red shaded region is the 95% CrI of fitted GP.

Prevalence estimates

Country Prevalence median (95% CrI) Total reported cases New reported cases (tallied over last 10 days) Population
Afghanistan 0.11% (0.05% - 0.26%) 22,890 9,231 38,928,341
Albania 0.023% (0.013% - 0.075%) 1,385 286 2,877,800
Algeria 0.027% (0.013% - 0.066%) 10,589 1,455 43,851,043
Andorra 0.69% (0.21% - 2.3%) 852 88 77,265
Argentina 0.12% (0.059% - 0.26%) 27,360 11,954 45,195,777
Armenia 0.77% (0.39% - 1.7%) 14,669 5,993 2,963,234
Australia 0.00097% (0.00057% - 0.0022%) 7,285 112 25,499,881
Austria 0.016% (0.0058% - 0.046%) 16,964 370 9,006,400
Azerbaijan 0.099% (0.051% - 0.23%) 8,882 3,893 10,139,175
Bahamas 0.00085% (0.00033% - 0.0049%) 103 1 393,248
Bahrain 0.82% (0.52% - 1.5%) 17,269 6,820 1,701,583
Bangladesh 0.055% (0.028% - 0.12%) 78,052 35,208 164,689,383
Belarus 0.24% (0.15% - 0.45%) 51,816 11,052 9,449,321
Belgium 0.086% (0.044% - 0.2%) 59,711 1,650 11,589,616
Bolivia 0.36% (0.19% - 0.77%) 16,165 7,434 11,673,029
Bosnia and Herzegovina 0.04% (0.015% - 0.16%) 2,831 346 3,280,815
Brazil 1.1% (0.61% - 2.3%) 802,828 337,662 212,559,409
Bulgaria 0.085% (0.04% - 0.22%) 3,086 587 6,948,445
Burkina Faso 0.00077% (0.00029% - 0.0029%) 892 45 20,903,278
Cameroon 0.026% (0.016% - 0.053%) 8,681 3,245 26,545,864
Canada 0.2% (0.11% - 0.42%) 97,519 8,112 37,742,157
Chad 0.0019% (0.00075% - 0.0083%) 848 89 16,425,859
Chile 0.71% (0.43% - 3.8%) 154,092 63,454 19,116,209
China 2.9e-05% (1e-05% - 0.00015%) 84,216 93 1,439,323,774
Colombia 0.19% (0.1% - 0.41%) 43,682 16,994 50,882,884
Congo 0.013% (0.0048% - 0.041%) 745 158 5,518,092
Costa Rica 0.024% (0.013% - 0.064%) 1,538 516 5,094,114
Côte d’Ivoire 0.014% (0.0081% - 0.029%) 4,404 1,654 26,378,275
Croatia 0.00089% (3e-04% - 0.0035%) 2,249 4 4,105,268
Cuba 0.0048% (0.0025% - 0.014%) 2,219 214 11,326,616
Cyprus 0.0068% (0.0037% - 0.021%) 975 34 1,207,361
Czechia 0.021% (0.0088% - 0.055%) 9,886 690 10,708,982
Democratic Republic of the Congo 0.005% (0.0024% - 0.015%) 4,514 1,681 89,561,404
Denmark 0.026% (0.012% - 0.069%) 12,035 442 5,792,203
Djibouti 0.33% (0.2% - 0.73%) 4,398 1,484 988,002
Dominican Republic 0.1% (0.059% - 0.22%) 21,437 4,906 10,847,904
Ecuador 0.25% (0.13% - 0.54%) 44,440 5,869 17,643,060
Egypt 0.091% (0.048% - 0.19%) 39,726 17,644 102,334,403
El Salvador 0.067% (0.031% - 0.16%) 3,481 1,203 6,486,201
Equatorial Guinea 0.042% (0.024% - 0.095%) 1,306 263 1,402,985
Estonia 0.041% (0.015% - 0.11%) 1,965 106 1,326,539
Ethiopia 0.0072% (0.0031% - 0.018%) 2,670 1,702 114,963,583
Finland 0.014% (0.0068% - 0.04%) 7,064 288 5,540,718
France 0.052% (0.028% - 0.11%) 155,561 5,893 65,273,512
Gabon 0.08% (0.05% - 0.16%) 3,463 850 2,225,728
Georgia 0.0054% (0.0028% - 0.016%) 831 85 3,989,175
Germany 0.03% (0.015% - 0.064%) 185,674 4,478 83,783,945
Ghana 0.018% (0.011% - 0.033%) 10,358 2,742 31,072,945
Greece 0.015% (0.0063% - 0.043%) 3,088 179 10,423,056
Guatemala 0.28% (0.13% - 0.66%) 8,561 3,954 17,915,567
Guinea 0.011% (0.0071% - 0.022%) 4,372 716 13,132,792
Guyana 0.0038% (0.0013% - 0.021%) 158 8 786,559
Haiti 0.049% (0.027% - 0.12%) 3,941 2,357 11,402,533
Honduras 0.19% (0.085% - 0.47%) 7,669 2,783 9,904,608
Hungary 0.029% (0.013% - 0.072%) 4,039 198 9,660,350
Iceland 0.0014% (0.00076% - 0.004%) 1,807 2 341,250
India 0.057% (0.031% - 0.12%) 297,535 123,772 1,380,004,385
Indonesia 0.033% (0.017% - 0.068%) 35,295 10,079 273,523,621
Iran 0.18% (0.094% - 0.36%) 180,176 33,508 83,992,953
Iraq 0.21% (0.11% - 0.45%) 16,675 10,802 40,222,503
Ireland 0.08% (0.034% - 0.22%) 25,238 362 4,937,796
Isle of Man 0% (0% - 0%) 336 0 85,032
Israel 0.049% (0.026% - 0.11%) 18,701 1,714 8,655,541
Italy 0.11% (0.057% - 0.22%) 236,142 3,894 60,461,828
Japan 0.0026% (0.0012% - 0.0065%) 17,332 528 126,476,458
Kazakhstan 0.038% (0.024% - 0.074%) 13,872 3,490 18,776,707
Kenya 0.011% (0.0048% - 0.029%) 3,215 1,470 53,771,300
Kosovo 0.042% (0.02% - 0.12%) 1,326 278 1,810,366
Kuwait 0.45% (0.28% - 0.82%) 34,432 9,248 4,270,563
Kyrgyzstan 0.016% (0.0088% - 0.041%) 2,166 444 6,524,191
Latvia 0.0068% (0.0025% - 0.023%) 1,094 30 1,886,202
Lebanon 0.0087% (0.0044% - 0.027%) 1,402 230 6,825,442
Liberia 0.021% (0.0059% - 0.081%) 410 137 5,057,677
Lithuania 0.023% (0.0096% - 0.076%) 1,752 90 2,722,291
Luxembourg 0.023% (0.01% - 0.061%) 4,052 40 625,976
Malaysia 0.0041% (0.0026% - 0.0082%) 8,369 637 32,365,998
Mali 0.023% (0.01% - 0.054%) 1,722 496 20,250,834
Mauritania 0.18% (0.072% - 0.48%) 1,162 739 4,649,660
Mauritius 5e-04% (2e-04% - 0.003%) 337 2 1,271,767
Mexico 0.94% (0.51% - 1.9%) 133,974 49,347 128,932,753
Moldova 0.44% (0.22% - 0.94%) 10,727 2,831 4,033,963
Morocco 0.0046% (0.0029% - 0.0099%) 8,537 823 36,910,558
Netherlands 0.075% (0.036% - 0.17%) 48,251 2,125 17,134,873
New Zealand 0% (0% - 0%) 1,154 0 4,822,233
Nicaragua 0.031% (0.014% - 0.18%) 1,464 705 6,624,554
Niger 0.00049% (0.00013% - 0.002%) 974 19 24,206,636
Nigeria 0.01% (0.0051% - 0.023%) 14,554 5,252 206,139,587
North Macedonia 0.63% (0.3% - 1.4%) 3,542 1,412 2,083,380
Norway 0.0092% (0.0044% - 0.039%) 8,594 183 5,421,242
Oman 0.41% (0.26% - 0.74%) 19,954 10,134 5,106,622
Pakistan 0.12% (0.063% - 0.24%) 125,933 59,476 220,892,331
Panama 0.76% (0.38% - 1.7%) 18,586 6,055 4,314,768
Paraguay 0.0094% (0.0057% - 0.02%) 1,230 313 7,132,530
Peru 0.93% (0.5% - 1.9%) 214,788 66,503 32,971,846
Philippines 0.018% (0.0099% - 0.038%) 24,175 7,541 109,581,085
Poland 0.067% (0.032% - 0.16%) 28,201 5,046 37,846,605
Portugal 0.19% (0.093% - 0.43%) 35,910 3,964 10,196,707
Puerto Rico 0.13% (0.077% - 0.28%) 5,352 1,705 2,860,840
Qatar 1.9% (1% - 10%) 75,071 22,164 2,881,060
Romania 0.085% (0.043% - 0.2%) 21,182 2,200 19,237,682
Russia 0.25% (0.15% - 0.49%) 502,436 114,813 145,934,460
San Marino 0.14% (0.077% - 0.59%) 691 20 33,938
Sao Tome and Principe 0.2% (0.1% - 0.62%) 639 176 219,161
Saudi Arabia 0.49% (0.25% - 1%) 116,021 34,255 34,813,867
Senegal 0.018% (0.01% - 0.041%) 4,759 1,330 16,743,930
Serbia 0.018% (0.011% - 0.042%) 12,102 748 8,737,370
Sierra Leone 0.01% (0.0044% - 0.032%) 1,085 256 7,976,985
Singapore 0.21% (0.12% - 0.62%) 39,387 5,527 5,850,343
Sint Maarten 0% (0% - 0%) 77 0 42,882
Slovakia 0.001% (5e-04% - 0.003%) 1,541 21 5,459,643
Slovenia 0.0076% (0.0025% - 0.023%) 1,488 15 2,078,932
Somalia 0.013% (0.006% - 0.037%) 2,513 685 15,893,219
South Africa 0.27% (0.15% - 0.56%) 58,568 29,328 59,308,690
South Korea 0.0029% (0.0014% - 0.0083%) 12,003 562 51,269,183
South Sudan 0.014% (0.0071% - 0.037%) 1,604 610 11,193,729
Spain 0.022% (0.012% - 0.097%) 242,707 3,479 46,754,783
Sri Lanka 0.0032% (0.0019% - 0.0069%) 1,877 319 21,413,250
Sudan 0.059% (0.027% - 0.14%) 6,730 2,209 43,849,269
Sweden 0.72% (0.38% - 1.5%) 48,288 11,812 10,099,270
Switzerland 0.016% (0.0074% - 0.039%) 30,961 216 8,654,618
Tajikistan 0.027% (0.017% - 0.049%) 4,834 1,271 9,537,642
Thailand 0.00018% (9.1e-05% - 0.00049%) 3,125 49 69,799,978
Togo 0.0028% (0.0015% - 0.0082%) 524 96 8,278,737
Tunisia 0.00052% (0.00018% - 0.0023%) 1,087 16 11,818,618
Turkey 0.035% (0.019% - 0.072%) 174,023 11,903 84,339,067
Ukraine 0.064% (0.031% - 0.15%) 29,070 5,866 43,733,759
United Arab Emirates 0.16% (0.1% - 0.3%) 40,986 7,816 9,890,400
United Kingdom 0.45% (0.24% - 0.92%) 291,409 20,187 67,886,004
United Republic of Tanzania 0% (0% - 0%) 509 0 59,734,213
United States of America 0.44% (0.24% - 0.89%) 2,023,347 276,260 331,002,647
Uruguay 0.0033% (0.0012% - 0.011%) 847 31 3,473,727
Uzbekistan 0.0081% (0.0051% - 0.016%) 4,819 1,306 33,469,199
Venezuela 0.011% (0.0066% - 0.023%) 2,814 1,445 28,435,943
Yemen 0.049% (0.022% - 0.11%) 591 304 29,825,968

Table 1: Estimates for the prevalence of COVID-19 in each country with greater than 10 deaths. We use the under-reporting estimates to adjust the reported case curves and tally these up over the last ten days as a proxy for prevalence. See Detailed Methods for more details.

Adjusted symptomatic case estimates

Figure 2: Estimated number of new symptomatic cases, calculated using our temporal under-reporting estimates. We adjust the reported case numbers each day - for each country with an under-reporting estimate - using our temporal under-reporting estimates to arrive at an estimate of the true number of symptomatic cases each day. The shaded blue region represents the 95% CrI, calcuated directly using the 95% CrI of the temporal under-reporting estimate.

Reported cases

Figure 3: Reported number of cases each day, pulled from the ECDC and plotted against time for comparison with our estimated true numbers of symptomatic cases each day, adjusted using our under-reporting estimates.

Current under-reporting estimates

Country Percentage of symptomatic cases reported (95% CI) Total cases Total deaths
Afghanistan 25% (19%-32%) 35,928 1,211
Albania 42% (27%-64%) 4,466 123
Algeria 57% (42%-74%) 25,484 1,124
Andorra 46% (18%-96%) 889 52
Angola 29% (17%-52%) 812 33
Argentina 53% (45%-61%) 141,887 2,617
Armenia 60% (45%-75%) 36,613 692
Australia 96% (83%-100%) 13,306 133
Austria 93% (74%-100%) 20,148 711
Azerbaijan 87% (73%-99%) 28,980 391
Bahamas 68% (24%-100%) 274 11
Bahrain 99% (94%-100%) 37,996 134
Bangladesh 94% (76%-100%) 216,110 2,801
Belarus 38% (25%-58%) 66,688 519
Belgium 76% (55%-95%) 64,754 9,812
Benin 63% (37%-94%) 1,694 34
Bolivia 34% (28%-40%) 65,252 2,407
Bosnia and Herzegovina 50% (34%-71%) 9,460 272
Brazil 41% (36%-46%) 2,287,475 84,082
Bulgaria 50% (37%-67%) 9,853 329
Burkina Faso 81% (40%-100%) 1,075 53
Cameroon 93% (63%-100%) 16,708 385
Canada 64% (50%-79%) 112,659 8,874
Cape Verde 88% (62%-100%) 2,190 21
Central African Republic 92% (65%-100%) 4,590 58
Chad 72% (16%-100%) 915 75
Chile 83% (28%-100%) 338,759 8,838
China 96% (41%-100%) 86,045 4,649
Colombia 31% (27%-35%) 226,373 7,688
Congo 84% (59%-100%) 3,038 51
Costa Rica 97% (86%-100%) 13,129 80
Cote dIvoire 99% (94%-100%) 15,001 93
Croatia 85% (56%-100%) 4,634 128
Cuba 86% (43%-100%) 2,466 87
Cyprus 84% (45%-100%) 1,045 19
Czechia 95% (76%-100%) 14,800 365
Democratic Republic of the Congo 91% (59%-100%) 8,719 200
Denmark 86% (60%-100%) 13,390 612
Djibouti 92% (69%-100%) 5,031 58
Dominican Republic 96% (87%-100%) 57,615 1,006
Ecuador 34% (29%-41%) 78,148 5,439
Egypt 28% (24%-33%) 90,413 4,480
El Salvador 44% (33%-58%) 13,377 372
Equatorial Guinea 95% (64%-100%) 3,071 51
Estonia 61% (29%-99%) 2,027 69
Eswatini 66% (40%-97%) 2,021 28
Ethiopia 58% (42%-75%) 11,524 188
Finland 88% (47%-100%) 7,372 328
France 56% (43%-70%) 179,398 30,182
Gabon 98% (89%-100%) 6,588 47
Georgia 83% (47%-100%) 1,104 16
Germany 98% (90%-100%) 204,183 9,111
Ghana 99% (95%-100%) 29,672 153
Greece 64% (33%-99%) 4,110 201
Guatemala 39% (32%-48%) 42,192 1,632
Guinea 97% (84%-100%) 6,806 42
Guinea Bissau 70% (42%-99%) 1,954 26
Guyana 39% (17%-79%) 351 19
Haiti 54% (30%-90%) 7,197 154
Honduras 44% (36%-53%) 36,902 1,011
Hungary 25% (13%-50%) 4,398 596
Iceland 86% (45%-100%) 1,841 10
India 53% (48%-60%) 1,287,945 30,601
Indonesia 19% (16%-22%) 93,657 4,576
Iran 15% (13%-17%) 284,034 15,074
Iraq 37% (31%-43%) 102,226 4,122
Ireland 3.9% (1.7%-9.3%) 25,826 1,763
Israel 99% (95%-100%) 57,982 442
Italy 25% (20%-32%) 245,338 35,092
Jamaica 84% (48%-100%) 816 10
Japan 99% (94%-100%) 27,956 992
Jordan 76% (36%-100%) 1,131 11
Kazakhstan 92% (44%-100%) 78,486 585
Kenya 73% (55%-92%) 15,601 263
Kosovo 36% (25%-49%) 6,467 158
Kuwait 99% (95%-100%) 61,872 421
Kyrgyzstan 90% (13%-100%) 31,247 1,211
Latvia 46% (23%-83%) 1,203 31
Lebanon 92% (66%-100%) 3,260 43
Liberia 37% (12%-76%) 1,117 71
Libya 40% (28%-59%) 2,314 56
Lithuania 41% (24%-78%) 1,960 80
Luxembourg 92% (71%-100%) 5,952 112
Madagascar 93% (76%-100%) 8,381 70
Malawi 33% (20%-47%) 3,386 79
Malaysia 93% (50%-100%) 8,840 123
Maldives 88% (61%-100%) 3,103 15
Mali 73% (41%-99%) 2,494 123
Mauritania 97% (77%-100%) 6,027 155
Mexico 12% (10%-14%) 370,712 41,908
Moldova 48% (39%-59%) 22,105 719
Montenegro 49% (32%-74%) 1,991 39
Morocco 79% (40%-100%) 18,264 292
Nepal 99% (94%-100%) 18,241 43
Netherlands 92% (72%-100%) 52,404 6,139
New Zealand 58% (25%-98%) 1,206 22
Nicaragua 53% (21%-99%) 3,439 108
Niger 57% (18%-100%) 1,124 69
Nigeria 79% (64%-94%) 38,948 833
North Macedonia 33% (26%-45%) 9,669 445
Norway 74% (34%-100%) 9,062 255
Oman 99% (95%-100%) 72,646 355
Pakistan 87% (75%-98%) 270,400 5,763
Palestine 97% (88%-100%) 11,882 70
Panama 48% (40%-57%) 56,817 1,209
Paraguay 94% (75%-100%) 4,113 36
Peru 4% (3.1%-3.9%) 371,096 17,654
Philippines 90% (81%-96%) 74,390 1,871
Poland 60% (46%-76%) 41,580 1,651
Portugal 98% (80%-100%) 49,379 1,705
Puerto Rico 98% (87%-100%) 13,473 188
Qatar 91% (51%-100%) 108,244 164
Romania 34% (28%-42%) 41,275 2,126
Russia 64% (58%-70%) 795,038 12,892
San Marino 80% (19%-100%) 716 42
Sao Tome and Principe 86% (50%-100%) 749 14
Saudi Arabia 74% (60%-89%) 260,394 2,635
Senegal 56% (38%-79%) 9,266 178
Serbia 52% (38%-68%) 22,443 508
Sierra Leone 79% (36%-100%) 1,752 66
Singapore 95% (74%-100%) 49,098 27
Slovakia 88% (63%-100%) 2,089 28
Slovenia 64% (32%-98%) 2,033 115
Somalia 92% (61%-100%) 3,171 93
South Africa 53% (46%-60%) 408,052 6,093
South Korea 90% (54%-100%) 13,979 298
South Sudan 70% (43%-97%) 2,239 45
Sri Lanka 96% (83%-100%) 2,753 11
Sudan 21% (14%-30%) 11,237 708
Suriname 68% (39%-99%) 1,234 23
Sweden 61% (44%-80%) 78,763 5,676
Switzerland 94% (77%-100%) 33,913 1,693
Syria 11% (6.4%-18%) 584 35
Tajikistan 97% (72%-100%) 7,060 58
Thailand 82% (49%-100%) 3,279 58
Togo 86% (52%-100%) 828 16
Tunisia 81% (41%-100%) 1,406 50
Turkey 89% (74%-100%) 223,315 5,563
Ukraine 56% (43%-71%) 62,823 1,571
United Arab Emirates 98% (86%-100%) 57,988 342
United Kingdom 17% (15%-20%) 297,146 45,554
United States of America 78% (63%-100%) 4,034,102 144,242
Uruguay 36% (17%-61%) 1,141 34
Uzbekistan 98% (91%-100%) 18,986 103
Venezuela 95% (82%-100%) 13,613 129
Yemen 12% (7%-21%) 1,654 461
Zambia 67% (39%-95%) 3,583 128
Zimbabwe 56% (32%-94%) 2,124 28

Table 2: Estimates for the proportion of symptomatic cases reported in different countries using cCFR estimates based on case and death timeseries data from the ECDC. Total cases and deaths in each country is also shown. Confidence intervals calculated using an exact binomial test with 95% significance.

Adjusting for outcome delay in CFR estimates

During an outbreak, the naive CFR (nCFR), i.e. the ratio of reported deaths date to reported cases to date, will underestimate the true CFR because the outcome (recovery or death) is not known for all cases [6]. We can therefore estimate the true denominator for the CFR (i.e. the number of cases with known outcomes) by accounting for the delay from confirmation-to-death [2].

We assumed the delay from confirmation-to-death followed the same distribution as estimated hospitalisation-to-death, based on data from the COVID-19 outbreak in Wuhan, China, between the 17th December 2019 and the 22th January 2020, accounting right-censoring in the data as a result of as-yet-unknown disease outcomes (Figure 1, panels A and B in [8]). The distribution used is a Lognormal fit, has a mean delay of 13 days and a standard deviation of 12.7 days [8].

To correct the CFR, we use the case and death incidence data to estimate the proportion of cases with known outcomes [2,7]:

\[ u_{t} = \frac{ \sum_{j = 0}^{t} c_{t-j} f_j}{c_t}, \]

where \(u_t\) represents the underestimation of the proportion of cases with known outcomes [2,6,7] and is used to scale the value of the cumulative number of cases in the denominator in the calculation of the cCFR, \(c_{t}\) is the daily case incidence at time, \(t\) and \(f_t\) is the proportion of cases with delay of \(t\) between confirmation and death.

Approximating the proportion of symptomatic cases reported

At this stage, raw estimates of the CFR of COVID-19 correcting for delay to outcome, but not under-reporting, have been calculated. These estimates range between 1% and 1.5% [2–4]. We assume a CFR of 1.4% (95% CrI: 1.2-1.7%), taken from a recent large study [4], as a baseline CFR. We use it to approximate the potential level of under-reporting in each country. Specifically, we perform the calculation \(\frac{1.4\%}{\text{cCFR}}\) of each country to estimate an approximate fraction of cases reported.

Temporal variation model fitting

We estimate the level of under-reporting on every day for each country that has had more than ten deaths. We then fit a Gaussian Process (GP) model using the library greta and greta.gp. The parameters we fit and their priors are the following: \[ \begin{aligned} &\sigma \sim \text{Log Normal(-1, 1)}: \quad &\text{Variance of the reporting kernel} \\ &\text{L} \sim \text{Log Normal(4, 0.5)}: \quad &\text{Lengthscale of the reporting kernel} \\ &\sigma_{\text{obs}} \sim \text{Truncated Normal(0, 0.5)}, \quad &\text{Variance of the obseration kernel, truncated at 0} \end{aligned} \] The kernel is split into two components: the reporting kernel \(R\), and the observation kernel \(O\). The reporting component has a standard squared-exponential form. For the observation component, we use an i.i.d. noise kernel to acccount for observation overdispersion, which can smooth out overly clumped death time-series. This is important as some countries have been known to report an unusually large number of deaths on a single day, due to past under-reporting.

In the sampling and fitting process, we calculate the expected number of deaths at each time-point, given the baseline CFR. We then use a Poisson likelihood, where the expected number of deaths is the rate of the Poisson likelihood, given the observed number of deaths

Approximating prevalence

We use the adjusted case curves, adjusted for under-reporting and for asymptomatic infections as a proxy for prevalence. Specifically, we tally up the adjusted new cases each day over the last ten days and calculate what percentage of the population in question this total equates to. This serves as a crude prevalence estimate. We assume ten days of infectiousness as taken from the mean of the total infectious period [9].

Adjusting case counts for under-reporting

We adjust the reported number of cases each day, pulled from the ECDC. Specifically, we divide the case numbers of each day by our “proportion of cases reported” estimates that we calculate each day for each country.*

Limitations

Implicit in assuming that the under-reporting is \(\frac{1.4\%}{\text{cCFR}}\) for a given country is that the deviation away from the assumed 1.4% CFR is entirely down to under-reporting. In reality, burden on healthcare system is a likely contributing factor to higher than 1.4% CFR estimates, along with many other country specific factors.

The following is a list of the other prominent assumptions made in our analysis:

Code and data availability

The code is publically available at https://github.com/thimotei/CFR_calculation. The data required for this analysis is a time-series for both cases and deaths, along with the corresponding delay distribution. We scrape this data from ECDC, using the NCoVUtils package [10].

The under-reporting estimates for all countries can be downloaded as a single .csv file here.

Similarly, global prevalence estimates can be downloaded as a single .csv file here

Acknowledgements

The authors, on behalf of the Centre for the Mathematical Modelling of Infectious Diseases (CMMID) COVID-19 working group, wish to thank DSTL for providing the High Performance Computing facilities and associated expertise that has enabled these models to be prepared, run and processed and in an appropriately-rapid and highly efficient manner.

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